Optimal. Leaf size=149 \[ \frac{5 c^{7/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right ),\frac{1}{2}\right )}{21 b^{9/4} \sqrt{b x^2+c x^4}}+\frac{10 c \sqrt{b x^2+c x^4}}{21 b^2 x^{5/2}}-\frac{2 \sqrt{b x^2+c x^4}}{7 b x^{9/2}} \]
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Rubi [A] time = 0.181635, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {2025, 2032, 329, 220} \[ \frac{5 c^{7/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{21 b^{9/4} \sqrt{b x^2+c x^4}}+\frac{10 c \sqrt{b x^2+c x^4}}{21 b^2 x^{5/2}}-\frac{2 \sqrt{b x^2+c x^4}}{7 b x^{9/2}} \]
Antiderivative was successfully verified.
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Rule 2025
Rule 2032
Rule 329
Rule 220
Rubi steps
\begin{align*} \int \frac{1}{x^{7/2} \sqrt{b x^2+c x^4}} \, dx &=-\frac{2 \sqrt{b x^2+c x^4}}{7 b x^{9/2}}-\frac{(5 c) \int \frac{1}{x^{3/2} \sqrt{b x^2+c x^4}} \, dx}{7 b}\\ &=-\frac{2 \sqrt{b x^2+c x^4}}{7 b x^{9/2}}+\frac{10 c \sqrt{b x^2+c x^4}}{21 b^2 x^{5/2}}+\frac{\left (5 c^2\right ) \int \frac{\sqrt{x}}{\sqrt{b x^2+c x^4}} \, dx}{21 b^2}\\ &=-\frac{2 \sqrt{b x^2+c x^4}}{7 b x^{9/2}}+\frac{10 c \sqrt{b x^2+c x^4}}{21 b^2 x^{5/2}}+\frac{\left (5 c^2 x \sqrt{b+c x^2}\right ) \int \frac{1}{\sqrt{x} \sqrt{b+c x^2}} \, dx}{21 b^2 \sqrt{b x^2+c x^4}}\\ &=-\frac{2 \sqrt{b x^2+c x^4}}{7 b x^{9/2}}+\frac{10 c \sqrt{b x^2+c x^4}}{21 b^2 x^{5/2}}+\frac{\left (10 c^2 x \sqrt{b+c x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{b+c x^4}} \, dx,x,\sqrt{x}\right )}{21 b^2 \sqrt{b x^2+c x^4}}\\ &=-\frac{2 \sqrt{b x^2+c x^4}}{7 b x^{9/2}}+\frac{10 c \sqrt{b x^2+c x^4}}{21 b^2 x^{5/2}}+\frac{5 c^{7/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{21 b^{9/4} \sqrt{b x^2+c x^4}}\\ \end{align*}
Mathematica [C] time = 0.0141234, size = 57, normalized size = 0.38 \[ -\frac{2 \sqrt{\frac{c x^2}{b}+1} \, _2F_1\left (-\frac{7}{4},\frac{1}{2};-\frac{3}{4};-\frac{c x^2}{b}\right )}{7 x^{5/2} \sqrt{x^2 \left (b+c x^2\right )}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.184, size = 134, normalized size = 0.9 \begin{align*}{\frac{1}{21\,{b}^{2}} \left ( 5\,\sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ) \sqrt{-bc}{x}^{3}c+10\,{c}^{2}{x}^{4}+4\,bc{x}^{2}-6\,{b}^{2} \right ){x}^{-{\frac{5}{2}}}{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{c x^{4} + b x^{2}} x^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{c x^{4} + b x^{2}} \sqrt{x}}{c x^{8} + b x^{6}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{\frac{7}{2}} \sqrt{x^{2} \left (b + c x^{2}\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{c x^{4} + b x^{2}} x^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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